{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using Bridge # needs master\n",
    "using StaticArrays\n",
    "SV = SVector{3,Float64} # A point in 3 dimensions\n",
    "SM = SMatrix{3,3,Float64,9}; # A 3x3 linear transformation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using Luxor, Colors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "srand(1234);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# centers of each of the three vortices (physically correct: whirlpool)\n",
    "z1, z2, z3 = zz = [SV(0,0,0), SV(-1,-1/sqrt(3),0), SV(-1, 1/sqrt(3), 0)]\n",
    "\n",
    "# x, y - space coordinates\n",
    "# z - age coordinate\n",
    "\n",
    "# drift acts on age (dots get older)\n",
    "β = SV(0, 0, 0.5)\n",
    "\n",
    "# mean reversion matrix (real eigenvalues correspond to forces of attraction, \n",
    "# complex eigenvalues to rotational forces)\n",
    "B = SM([-0.2  1   0 \n",
    "        -1   -0.2 0 \n",
    "         0    0   0]);\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "b (generic function with 1 method)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# according to age, points are attracted to different vortices\n",
    "b(t, x) = B*(x - (x[3]<10)*z1 - (10<x[3]<20)*z2 - (20< x[3]<30)*z3) + β"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "6001"
      ],
      "text/plain": [
       "6001"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# time axis\n",
    "tt = collect(0:0.01:60); length(tt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element SVector{3,Float64}:\n",
       " -1.0    \n",
       "  0.57735\n",
       "  0.0    "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# starting point in third vortex\n",
    "u = z3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Bridge.SamplePath{SVector{3,Float64}}([0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09  …  59.91, 59.92, 59.93, 59.94, 59.95, 59.96, 59.97, 59.98, 59.99, 60.0], SVector{3,Float64}[[-1.0, 0.57735, 0.0], [-0.99219, 0.586148, 0.005], [-0.984309, 0.594849, 0.01], [-0.976356, 0.603454, 0.015], [-0.968335, 0.611962, 0.02], [-0.960244, 0.620373, 0.025], [-0.952086, 0.628686, 0.03], [-0.943862, 0.6369, 0.035], [-0.935573, 0.645015, 0.04], [-0.92722, 0.653031, 0.045]  …  [-1.01878, 0.566425, 29.955], [-1.01885, 0.566634, 29.96], [-1.01892, 0.566845, 29.965], [-1.01899, 0.567055, 29.97], [-1.01905, 0.567265, 29.975], [-1.01911, 0.567476, 29.98], [-1.01917, 0.567687, 29.985], [-1.01923, 0.567898, 29.99], [-1.01928, 0.56811, 29.995], [-1.01934, 0.568321, 30.0]]), 0.006184038207458661)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# allocate space for solution `U` (both time axis and locations)\n",
    "U = SamplePath(tt, zeros(SV, length(tt)))\n",
    "\n",
    "# solve dynamical system with derivative `b`, start in `u` and third order method BS3\n",
    "solve!(Bridge.BS3(), U, u, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "colorf (generic function with 1 method)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# color points according to age\n",
    "function colorf(x,y,z) \n",
    "    j = 10<z<20 ? 2 : 20< z< 30 ? 3 : 1\n",
    "    [colorant\"brown3\", colorant\"forestgreen\", colorant\"mediumorchid3\"][j]\n",
    "end    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# show the solution\n",
    "@svg begin\n",
    "    for p in U.yy[1:10:end]\n",
    "        (x,y,z) = p\n",
    "        g = Point(100y,-100x) \n",
    "        sethue(colorf(x,y,z))\n",
    "        circle(g, 4, :fill)\n",
    "    end    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define stochastic differential equation\n",
    "#\n",
    "# Dynamical system with diffusivity corresponding to additive \n",
    "# Gaussian noise with variance '(0.1)^2' per unit time and dynamics 'b'\n",
    "struct LogoP <: ContinuousTimeProcess{SV}\n",
    "end\n",
    "Bridge.b(s, x, ::LogoP) = b(s, x) # drift\n",
    "Bridge.σ(s, x, ::LogoP) = 0.1I\n",
    "Bridge.a(s, x, ::LogoP) = 0.1*0.1*I # diffusivity\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# allocate path of solution\n",
    "X = copy(U);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# solve the stochastic differential equation with `EulerMaruyama`\n",
    "solve!(Bridge.EulerMaruyama(), X, u, sample(tt, Wiener{SV}()), LogoP() );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot `X`. Note the randomness\n",
    "@svg begin\n",
    "    for p in X.yy[1:10:end]\n",
    "        (x,y,z) = p\n",
    "        g = Point(100y,-100x) \n",
    "        sethue(colorf(x,y,z))\n",
    "        circle(g, 4, :fill)\n",
    "    end    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define a linear approximation to the stochastic differential equation\n",
    "struct LogoAppr <: ContinuousTimeProcess{SV}\n",
    "    t0 # starting time\n",
    "    z0 # starting age\n",
    "end\n",
    "b2(t, x) = B*x + Bridge.β(t, LogoAppr(0.0, u[3])) \n",
    "Bridge.B(t, ::LogoAppr) = B\n",
    "\n",
    "# age (which growth randomly) substituted by deterministic time make\n",
    "# the system linear\n",
    "function Bridge.β(t, P::LogoAppr) \n",
    "    z = P.z0 + (t - P.t0)*β[3]\n",
    "    B*( - (z<10)*z1 - (10<z<20)*z2 - (20< z <30)*z3) + β\n",
    "end\n",
    "\n",
    "# same diffusivity\n",
    "Bridge.a(t, ::LogoAppr) = Bridge.a(t, u, LogoP()) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
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       "</svg>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Without noise age growths deterministically and LogoAppr has the same \n",
    "# dynamics as LogoP\n",
    "\n",
    "U2 = SamplePath(tt, zeros(SV, length(tt)));\n",
    "solve!(Bridge.BS3(), U2, u, b2)\n",
    "@svg begin\n",
    "    for p in U2.yy[1:10:end]\n",
    "        (x,y,z) = p\n",
    "        g = Point(100y,-100x) \n",
    "        sethue(colorf(x,y,z))\n",
    "        circle(g, 4, :fill)\n",
    "    end    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# pick some time span for animation\n",
    "n0 = 100\n",
    " m = 59\n",
    "nend = n0 + m - 1\n",
    "frames = n0:nend;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "m2 = 20 # subsampling factor\n",
    "\n",
    "# now we define the time axis of the animation\n",
    "tt2 = collect(linspace(tt[n0], tt[nend], m*m2));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# number of particles\n",
    "K = 1000\n",
    "\n",
    "# reserve space\n",
    "XX0 = [SamplePath(tt[1:n0], zeros(SV, length(tt[1:n0]))) for k in 1:K];\n",
    "XX = [SamplePath(tt2, zeros(SV, length(tt2))) for k in 1:K];\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# evolve XX0 starting in U for n0*dt steps (randomizes U somewhat)\n",
    "for k in 1:K\n",
    "    solve!(Bridge.EulerMaruyama(), XX0[k], U.yy[length(tt)÷K*k], sample(tt[1:n0], Wiener{SV}()), LogoP() )\n",
    "end\n",
    "# take positions of particles at end-time of `XX0` as starting points of `XX` \n",
    "us = [XX0[k].yy[end] for k in 1:K] \n",
    "for k in 1:K\n",
    "    # CONSTRAINED/CONDITIONAL stochastic differential equation: particle `k` moves to starting position of particle `k + 11`\n",
    "    # Why 11? the particle `k+11` is on average as old now as the particle `k` will be at the end of `tt2`\n",
    "    GP = Bridge.GuidedBridge(tt2, (us[k], us[mod1(k+11,K)]), LogoP(), LogoAppr(tt2[1], us[k][3]))\n",
    "    \n",
    "    # solve the CONSTRAINED/CONDITIONAL stochastic differential equation\n",
    "    Bridge.bridge!(Bridge.EulerMaruyama(), XX[k], sample(tt2, Wiener{SV}()), GP)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a path looks is constrained to start in us[k] and end in us[k+1]\n",
    "k = 3; GP = Bridge.GuidedBridge(tt2, (XX0[k].yy[1], XX0[mod1(k+11,K)].yy[1]), LogoP(), LogoAppr(tt2[1], XX0[k].yy[1][3]))\n",
    "GP.tt;\n",
    "@svg begin\n",
    "    for p in XX[k].yy[1:m2:end]\n",
    "        (x,y,z) = p\n",
    "        g = Point(100y,-100x) \n",
    "        sethue(colorf(x,y,z))\n",
    "        circle(g, 1, :fill)\n",
    "    end    \n",
    "    sethue(colorant\"black\")\n",
    "    for p in [us[k], us[mod1(k+11,K)]]\n",
    "        (x,y,z) = p\n",
    "        g = Point(100y,-100x) \n",
    "        circle(g, 1, :fill)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "frame (generic function with 1 method)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define movie frame\n",
    "function frame(szene, i)\n",
    "    for k in 1:K\n",
    "        x, y, z = XX[k].yy[1 + (i-1)*m2] #subsample\n",
    "        g = Point(50y,-35-50x) \n",
    "        sethue(colorf(x,y,z))\n",
    "        circle(g, 2, :fill)\n",
    "    end\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Luxor.Movie(175.0, 175.0, \"test\", 1:59)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# build movie\n",
    "bang = Movie(350÷2, 350÷2, \"test\", 1:m)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "backdrop (generic function with 1 method)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "backdrop(scene, framenumber) =  background(\"white\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mFrames for animation \"test\" are being stored in directory: \n",
      "\t /var/folders/zm/_k619j950rgcj72mlr4phfgjyj_8r5/T/tmpDWK8f1\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36m... 59 frames saved in directory:\n",
      "\t /var/folders/zm/_k619j950rgcj72mlr4phfgjyj_8r5/T/tmpDWK8f1\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mGIF is: /tmp/animationtest.gif\n",
      "\u001b[39m"
     ]
    },
    {
     "data": {
      "text/html": [
       "true"
      ],
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create movie: see output for gif location\n",
    "animate(bang, [Scene(bang, backdrop, 1:m), Scene(bang, frame, 1:m)], framerate=10, creategif = true, pathname=\"/tmp/animationtest.gif\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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   "mimetype": "application/julia",
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